A homological model for $U_q \mathfrak{sl}(2)$ Verma-modules and their braid representations

DOI10.2140/GT.2022.26.1225arXiv2002.08785MaRDI QIDQ6335224

Jules Martel

Publication date: 20 February 2020

Abstract: We extend Lawrence's representations of the braid groups to relative homology modules, and we show that they are free modules over a Laurent polynomials ring. We define homological operators and we show that they actually provide a representation for an integral version for $U_{q}mathfraksl(2)$. We suggest an isomorphism between a given basis of homological modules and the standard basis of tensor products of Verma modules, and we show it to preserve the integral ring of coefficients, the action of $U_{q}mathfraksl(2)$, the braid group representations and their grading. This recovers an integral version for Kohno's theorem relating absolute Lawrence representations with quantum braid representation on highest weight vectors. It is an extension of the latter theorem as we get rid of generic conditions on parameters, and as we recover the entire product of Verma-modules as a braid group and a $U_{q}mathfraksl(2)$-module.

Mathematics Subject Classification ID

Covering spaces and low-dimensional topology (57M10) Quantum groups (quantized enveloping algebras) and related deformations (17B37) Braid groups; Artin groups (20F36) Topological quantum field theories (aspects of differential topology) (57R56) Discriminantal varieties and configuration spaces in algebraic topology (55R80) Homology with local coefficients, equivariant cohomology (55N25)

This page was built for publication: A homological model for $U_q \mathfrak{sl}(2)$ Verma-modules and their braid representations

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q6335224)